In all exercises, other than exercises with no solution, use i...

Question

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. (x - 4)/6 ≥ (x - 2)/9 + 5/18

Accepted Answer

Hello today we're going to be solving the following inequality and we're going to be graphing its solution and writing the solution in interval notation. So what we are given is x minus 3/7 is greater than or equal to x minus 2/8 plus 1/28. Now we want to go ahead and first get rid of the denominators given to us. And in order to do that we can go ahead and multiply each fraction by the lowest common denominator of the three given denominators. So what we are given is 78 and 28. And the lowest common multiple or the lowest common denominator between the three is going to be 56. So what we can go ahead and do is multiplying every term by 56. That way we can go ahead and simplify the given inequality, multiplying every term by 56 is going to give us eight times the quantity x minus three is greater than or equal to seven times the quantity x minus two plus two times one. Now, in order to simplify this inequality further, we're going to distribute eight into the quantity x minus three and we're going to distribute seven into the quantity X minus two. Doing this is going to give us eight X minus 24 is greater than or equal to seven x minus 14 plus two times one which is going to be too. And on the right hand side we can go ahead and combine negative 14 with two to give us -12. All right now, what we want to go ahead and do is get all of the X terms to one side of the inequality and all the constants to the other side of the inequality. And in order to do this, we can go ahead and subtract seven X to both sides of the inequality And add 24 to both sides. And doing this is going to give us eight x minus seven X. Which is going to give us just X is greater than or equal to negative 12 plus 24 which is positive 12. So this is the inequality in its most simplified form. Now, what we want to go ahead and do is graph this inequality. So what we start to do is draw a number line And we're going to go ahead and put the number 12 in the middle of this number line. Now that inequality says that we want a number X that is greater than or equal to 12. So we want all the values of X that are going to be greater than 12. So in other words, we want all the positive values going to positive infinity and the number X can be equal to 12 itself. So we're going to include 12 in our solution set and we can represent that with a solid line. So we're including 12 and we want all the numbers going to infinity. So we're going to shade in the whole right hand side of 12. And in order to put this an interval notation, we first start off by writing 12 are least number in the solution set. And since we're including 12 in the solution set, we're going to assign a bracket to it. A bracket indicates that the value is being included in our solution set And we want all the values going to positive infinity. So the interval notation is going to be bracket 12 comma infinity. And we're going to close it off with the parentheses and that's going to be the solution to this inequality. And the graph that accurately represents the solution is going to be C. So C is going to be the answer to this problem. So I hope this video helps you in approaching this inequality problem and I'll go ahead and see you all in the next video.

In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. (x - 4)/6 ≥ (x - 2)/9 + 5/18

Key Concepts

Solving Linear Inequalities

Manipulating Fractions in Inequalities

Interval Notation and Graphing Solutions

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